Inverse design of 3D reconfigurable curvilinear modular origami structures using geometric and topological reconstructions

The recent development of modular origami structures has ushered in an era for active metamaterials with multiple degrees of freedom (multi-DOF). Notably, no systematic inverse design approach for 3D curvilinear modular origami structures has been reported. Moreover, very few modular origami topologies have been studied to design active metamaterials with multi-DOF. Herein, we develop an inverse design method for constructing 3D reconfigurable architected structures — we synthesize modular origami structures whose unit cells can be volumetrically mapped into a prescribed 3D curvilinear shape followed by volumetric shrinkage to construct modules. After modification of the tubular geometry, we search through all the possible geometric and topological combinations of the modular origami structures to attain the target mobility using a topological reconstruction of modules. Our inverse design using geometric and topological reconstructions can provide an effective solution to construct 3D curvilinear reconfigurable structures with multi-DOF. Our work opens a path toward 3D reconfigurable systems based on volumetric inverse design, such as 3D active metamaterials and 3D morphing devices for automotive, aerospace, and biomedical engineering applications.


Supplementary Note 1: Geometric reconstruction Construction of 2D modular origami structures
Our geometric reconstruction primarily used for 3D curvilinear architected structures in the main text can also be applied to planar cases, as illustrated in Supplementary Figure 1. A unit cell can be mapped to a target planar geometry followed by planar shrinkage. One can extrude tubes parallel to the lines connecting adjacent centroids of the shrunken polyhedrons to make origami modules, connecting with adjacent modules to construct an assembly.
Supplementary Figure 1. Geometric reconstruction of a planar 2D curvilinear architected structure with a 2 × 2 × 1 mapping of a unit cell.

Size dependency on mapping
We quantify the effect of mapping density on the volumetric change of a unit cell. We fill a sphere by mapping a cubic unit cell with different system sizes, as shown in Supplementary Figure 2a. The selected polyhedron, which is commonly close to the boundary of the template, undergoes a severe volume change as the system size increases, as illustrated in Supplementary Figure 2b. The volume change ( 0 − ) 0 ⁄ of the polyhedron continuously increases from 49.9% to 84.3% as the mapping density increases from 2 × 2 × 2 to 6 × 6 × 6, where 0 and are the volumes of the polyhedron before and after mapping, respectively.
We also quantify the effect of the mapping density on the filling efficiency, as indicated by the volumetric ratio 0 ⁄ in Supplementary Figure 2c. g and 0 are the volumes of the template and the target shape, respectively. Although it is apparent that a denser mapping approximates the targeting shape with higher accuracy, we notice a specific system size may balance the accuracy and computational cost. In this example with a spherical target shape, a 3 × 3 × 3 system provides a good filling accuracy of 0 ⁄ (= 90.62%), close to the filling accuracy of a 4 × 4 × 4 system with 0 ⁄ (= 93.76%). Note that the 3 × 3 × 3 system uses a much smaller number of polyhedrons (27) than the 4 × 4 × 4 system (64),

Supplementary Note 2: Inverse design of reconfigurability
After the geometric reconstruction, the modules constructed from the shrunken polyhedrons generally produce immobility due to the irregular polygon tubes; the spatially connected irregular polygon tubes produce immobility. Some modules consist of a combination of regular and irregular polygon tubes, partially producing mobility. However, the mobility of the flexible modules is damaged when connected with adjacent rigid modules, as illustrated in Supplementary

Foldability conditions
The foldability constraints in Equations (4) and (5) in the main text, • = and • = , vary depending on the shape of the polygons where prismatic tubes are extruded. Supplementary Figure 5 presents examples of the form of the foldability constraints for varying polygons. We focus on the tubes with parallel hinges to explain the form of • = in these examples; however, the principles of applying constraints are the same for the tubes with intersecting hinges, where • = .
For instance, in the case in Supplementary Figure 5a.1 with the strictest constraints on foldability, we have The quadrilateral tubes satisfying these constraints have a rhombic cross-section, reaching its maximum sole motion for flat-foldability along any folding path.
For the tube in Supplementary Figure 5b.1 with an odd-number face in a pentagon shape, The pentagonal tube reaches its maximum sole motion for flat-foldability along two possible folding paths.
In a single tube, a higher number of foldability constraints increase the range of motion. In the example where = 4, the quadrilateral tube with a higher number of constraints is flat-foldable along any path, as shown in Supplementary Figures 5a.1 and a.2. However, the tube with one constraint can only be flatfolded along one path, as shown in Supplementary Figure 5a.3.
In this study, we applied the strictest foldability constraints possible, with the maximum number of constraints ( or ) for each tube -the cases of Supplementary Figure 5a.1, b.1, and c.1. However, when the targeting shape is complicated, it may be challenging for the algorithm to find a solution while simultaneously satisfying the maximum foldability of the tubes and other constraints during geometric modification, e.g., persevering the targeting shape and enforcing planar plates, as described in Equations (3) and (6) of the main text. In this context, we apply a weaker foldability constraint such as that in Supplementary Figures 5a.2 foldability constraints in the hexagonal tube with = 6. In this study, the tubes extruded from the triangular faces could be exempted from geometrical modification due to their immobility.

Implementation
Note that the prismatic architected structures produced by the geometric reconstruction serve as an initial guess for the design of reconfigurable structures. As a preprocess of reconfigurability, we proceed with a geometric modification by solving the optimization problem in Equations (3) to (6) in the main text using the optimization tool fmincon in MATLAB.
The objective function, foldability, and the planar-face constraints in Equations (3)-(6) in the main text are functions of nodal position vectors ( 1 , 2 , 3 , … , ) of the architected materials with full nodes . We also specify gradient information for both the objective function and constraints using options of SpecifyObjectiveGradient and SpecifyConstraintGradient in fmincon.

The structural integrity of topological reconstruction
We apply the constraints on the graph structure in Equations (9) and (10)

Validation of top-down approach with 2D motion structures
To validate our inverse design with the kinematic mobility analysis (bottom-up approach), we present simple assemblies of 2D bar mechanisms. While removing links (bars) from a 3 × 3 linkage in Supplementary Figures 7a and 7b, we obtain mobility using Kutzbach's modification of Gruebler's mobility equation: where is the mobility, is the number of links, 1 is the number of full joints, and is the number of multiple joints counting as one less than the number of links joined at the point. Note that the nodes and edges in Supplementary Figures 7a and 7b represent hinges and binary links, respectively.
We compare the kinematic analysis with our inverse design via the topological reconstruction. Similar to Equations (7)-(10) in the main text, we formulate the design problem for target mobility ̅ : . . ∈ [0,1] for = 1,2,3, … , − ( ) + 2 ≤ 0, for = 1,2,3, … , where the binary design parameter represents the existence of the -th link, with 0 representing the links removed. and are the total number of links and hinges before removal, respectively.

Supplementary Note 3: Mobility analysis and simulation of transformation
To identify the DOFs of reconfigurable modular origami comprising rigid faces and flexible hinges, we calculate the number of free variables associated with the linearized constraint matrix 1 . We triangulate each face of the structure to facilitate the description of geometric constraints, for which an additional diagonal edge is generated on the face, as shown in Supplementary Figure 8a where = 1,2, and 3 . , is the displacement of the -th vertex along the -th axis of a Cartesian coordinate, = 1, … . is the total number of selected hinges, and = 1,2,3 … , . is the total number of vertices in the structure. A detailed derivation of ℎ can be found elsewhere 2 .
Next, we find the free column in the reduced row echelon form of . The free column corresponds to free variables in [ ]. All the free variables can be found in , i.e., the independent angles.
We obtain the transformed configurations of the reconfigurable architected materials using a numerical iterative method that applies a projection matrix to reduce numerical errors 3

Rigid structures
The geometry reconstruction using the volumetric mapping and shrinkage in the main text provides stiff and immobile structures suitable for lightweight structural applications. We prototype a rigid model using multi-material inkjet 3D printing (MultiJet, ProJet MJP 5600, 3D systems). To generate an input file for 3D printing, we write a MATLAB script converting a thin-walled model (Supplementary Figure 9a) into one with a prescribed wall thickness (Supplementary Figure 9b). For the multiJet printing, the thicknesses of the plate and hinges ℎ can be close (0.7 ≤ ℎ ≤ ), as shown in Supplementary Figure 9c. We assign VisiJet CR-CL (~1.3GPa ) to the faces and VisiJet CE-BK (~0.3 ) to the hinges of the model. Supplementary Figure 9d shows a printed prototype.

Reconfigurable structures
The geometric modification and topological reconstruction in the main text can build reconfigurable structures. We fabricate reconfigurable prototypes using i) assembly of patterned paperboards and ii) additive manufacturing by stereolithography (SLA) using Form 3 (Formlabs). For the assembly of a thinwalled model in Supplementary Figure 10a, we generate a papercutting path on paperboards (Supplementary Figure 10b) using an in-house pattern cutting machine, followed by connecting hinges with transparent tapes, as shown in Supplementary Figure 10c. Seeking fully automatic manufacturing, we utilize additive manufacturing with a single material. We set 0.05 ≤ ℎ ≤ 0.5 and assign a single soft material (Flexible 80A with ~2.3 MPa) for the entire model, as shown in Supplementary Figures 10d  and 10e. Supplementary Figure 10f shows a printed reconfigurable prototype.

Magnetic actuation
To demonstrate a reconfigurable structure (Supplementary Figure 11a) remotely controllable, we printed a hyperboloid prismatic modular origami using the topological reconstruction of tetrahedron modules and selectively embedded six permanent magnets (NdFeB) into triangular tubes, as shown in Supplementary  Figures 11c and 11d. We used ethyl vinyl acetate (EVA) hot glue to bond the magnets to the printed prototypes; the location of the magnets and their remnant flux orientation are shown in Supplementary  Figures 11c and 11d.
We apply a rotational uniform magnetic field to actuate the 3D printed prototype. The uniform magnetic field is generated by a Halbach array composed of a circumferential array of permanent magnets, where the cylindrical magnetic space has diameter (= 104 ) and height ℎ (= 26 ) , as shown in Supplementary Figure 11b. A uniform magnetic field is applied in the radial direction of the cylindrical hole with a magnitude of 80 mT.
Supplementary Figure 11. A magnetically triggered reconfigurable structure: (a) hyperboloid reconfigurable modular origami structure with a spatially gradient tessellation of a tetrahedron reference unit followed by geometric modification and topological reconstruction; (b) the Halbach array provides a rotational uniform magnetic field; (c) locations of permanent magnets embedded in the prototype; (d) remanent flux orientation of permanent magnets in the hyperboloid prototype.

Validation of motion with experiment
To validate our numerical algorithm of inverse design of reconfigurability, we fabricated one reconfigurable structure -the spherical structure with tetrahedron and octahedron modules of Figure 3b in the main text. Note that the reconfigurable structure has 10 DOFs after the topological reconstruction, as shown in Figure 3a7 of the main text. We used a 3D scanner (Einscan Pro) to capture the transformed shapes of an SLA 3D printed prototype. We compared the 10 dihedral angles of the reconfigurable structure during transformation obtained by numerical simulation and 3D scanned measurement, as shown in Supplementary Figure 12. Our model reasonably matches the experiment. The measured configurations were close to the simulated ones, as shown in Supplementary Figures 12a-12d. The measurement shows an average mismatch of ~3°≤ 1 ∑|∆ | ≤ 13° depending on the configurations and angles measured. Note that it is challenging to match our model with the experiment due to the comprehensive and accumulated errors such as calibration errors by the perspective 3D scanning view, hidden parts of interior connections during 3D scanning, dimensional errors during 3D printing, and so on.